Calculus

Precalculus: Graphical, Numerical, Algebraic | 8th Edition | ISBN: 9780321656933 | Authors: Franklin Demana, Bert K. Waits, Gregory D. Foley, Daniel Kennedy, Dave Bock
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Author: Franklin Demana, Bert K. Waits, Gregory D. Foley, Daniel Kennedy, Dave Bock
Publisher: Pearson
ISBN: 9780321656933
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Chapter 6 Problems

Chapter: 6 Problem: 6.1.1.401

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1 and 2, write the roots of the equation in a + bi form. \(x^{2}+13=4 x \)

Chapter: 6 Problem: 6.1.1.402

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1 and 2, write the roots of the equation in a + bi form. \(5\left(x^{2}+1\right)=6 x\)

Chapter: 6 Problem: 6.1.1.403

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3 and 4, write the complex number in standard form a + bi \((1+i)^{5}\)

Chapter: 6 Problem: 6.1.1.404

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3 and 4, write the complex number in standard form a + bi \((1-i)^{4}\)

Chapter: 6 Problem: 6.1.1.405

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=\frac{1}{2} \quad \text { and } \quad \cos \theta=-\frac{\sqrt{3}}{2}\)

Chapter: 6 Problem: 6.1.1.406

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=-\frac{\sqrt{2}}{2} \quad \text { and } \quad \cos \theta=\frac{\sqrt{2}}{2}\)

Chapter: 6 Problem: 6.1.1.407

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=-\frac{\sqrt{3}}{2} \quad \text { and } \quad \cos \theta=-\frac{1}{2}\)

Chapter: 6 Problem: 6.1.1.408

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=-\frac{\sqrt{2}}{2} \text { and } \cos \theta=-\frac{\sqrt{2}}{2}\)

Chapter: 6 Problem: 6.1.1.409

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 9 and 10, find all real solutions. \(x^{3}-1=0\)

Chapter: 6 Problem: 6.1.1.410

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 9 and 10, find all real solutions. \(x^{4}-1=0\)

Chapter: 6 Problem: 6.1.1.411

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1 and 2, plot all four points in the same complex plan. \(1+2 i, 3-i,-2+2 i, i\)

Chapter: 6 Problem: 6.1.1.412

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1 and 2, write the roots of the equation in \(a+b i\) form. \(x^{2}+13=4 x\)

Chapter: 6 Problem: 6.1.1.413

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3 and 4, write the complex number in standard form \(a+b i\). \((1+i)^{5}\)

Chapter: 6 Problem: 6.1.1.414

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3 and 4, write the complex number in standard form \(a+b i\). \((1+i)^{4}\)

Chapter: 6 Problem: 6.1.1.415

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=\frac{1}{2} \quad \text { and } \quad \cos \theta=-\frac{\sqrt{3}}{2}\)

Chapter: 6 Problem: 6.1.1.416

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=-\frac{\sqrt{2}}{2} \quad \text { and } \quad \cos \theta=\frac{\sqrt{2}}{2}\)

Chapter: 6 Problem: 6.1.1.417

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=-\frac{\sqrt{3}}{2} \text { and } \cos \theta=-\frac{1}{2}\)

Chapter: 6 Problem: 6.1.1.418

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 58, find an angle \(\theta \text { in } 0 \leq \theta<2 \pi\) that satisfies both equations. \(\sin \theta=-\frac{\sqrt{2}}{2} \text { and } \cos \theta=-\frac{\sqrt{2}}{2}\)

Chapter: 6 Problem: 6.1.1.419

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 9 and 10, find all real solutions. \(x^{3}-1=0\)

Chapter: 6 Problem: 6.1.1.420

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 9 and 10, find all real solutions. \(x^{4}-1=0\)

Chapter: 6 Problem: 6.1.1.421

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1 and 2, plot all four points in the same complex plan. \(1+2 i, 3-i,-2+2 i, i\)

Chapter: 6 Problem: 6.1.1.422

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1 and 2, plot all four points in the same complex plan. \(2-3 i, 1+i, 3,-2-i\)

Chapter: 6 Problem: 6.1.1.423

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(3 i\)

Chapter: 6 Problem: 6.1.1.424

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(-2 i\)

Chapter: 6 Problem: 6.1.1.425

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(2+2 i\)

Chapter: 6 Problem: 6.1.1.426

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(\sqrt{3}+i\)

Chapter: 6 Problem: 6.1.1.427

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(-2+2 i \sqrt{3}\)

Chapter: 6 Problem: 6.1.1.428

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(3-3 i\)

Chapter: 6 Problem: 6.1.1.429

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(3+2 i\)

Chapter: 6 Problem: 6.1.1.430

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\). \(4-7 i\)

Chapter: 6 Problem: 6.1.1.431

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\).

Chapter: 6 Problem: 6.1.1.432

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 312, find the trigonometric form of the complex number where the argument satisfies \(0 \leq \theta<2 \pi\).

Chapter: 6 Problem: 6.1.1.433

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1318, write the complex number in standard form \(a+b i\). \(3\left(\cos 30^{\circ}-i \sin 30^{\circ}\right)\)

Chapter: 6 Problem: 6.1.1.434

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 1318, write the complex number in standard form \(a+b i\). \(8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)\)

Chapter: 6 Problem: 6.1.1.435

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 2326, find the trigonometric form of the quotient. \(\frac{6(\cos 5 \pi+i \sin 5 \pi)}{3(\cos 2 \pi+i \sin 2 \pi)}\)

Chapter: 6 Problem: 6.1.1.436

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 2326, find the trigonometric form of the quotient. \(\frac{\cos (\pi / 2)+i \sin (\pi / 2)}{\cos (\pi / 4)+i \sin (\pi / 4)}\)

Chapter: 6 Problem: 6.1.1.437

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 2730, find the product \(z_{1} \cdot z_{2}\) and quotient \(z_{1} / z_{2}\) in two ways, (a) using the trigonometric form for \(z_{1} \text { and } z_{2}\) and

Chapter: 6 Problem: 6.1.1.438

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 2730, find the product \(z_{1} \cdot z_{2}\) and quotient \(z_{1} / z_{2}\) in two ways, (a) using the trigonometric form for \(z_{1} \text { and } z_{2}\) and

Chapter: 6 Problem: 6.1.1.439

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 2730, find the product \(z_{1} \cdot z_{2}\) and quotient \(z_{1} / z_{2}\) in two ways, (a) using the trigonometric form for \(z_{1} \text { and } z_{2}\) and

Chapter: 6 Problem: 6.1.1.440

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 2730, find the product \(z_{1} \cdot z_{2}\) and quotient \(z_{1} / z_{2}\) in two ways, (a) using the trigonometric form for \(z_{1} \text { and } z_{2}\) and

Chapter: 6 Problem: 6.1.1.441

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \(\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)^{3}\)

Chapter: 6 Problem: 6.1.1.442

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \(\left[3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)\right]^{5}\)

Chapter: 6 Problem: 6.1.1.443

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \(\left[2\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right]^{3}\)

Chapter: 6 Problem: 6.1.1.444

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \(\left[6\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)\right]^{4}\)

Chapter: 6 Problem: 6.1.1.445

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \((1+i)^{5}\)

Chapter: 6 Problem: 6.1.1.446

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \((3+4 i)^{20}\)

Chapter: 6 Problem: 6.1.1.447

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \((1-\sqrt{3} i)^{3}\)

Chapter: 6 Problem: 6.1.1.448

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3138, use De Moivres Theorem to find the indicated power of the complex number. Write your answer in standard form \(a+b i\) and support with a calculator. \(\left(\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{3}\)

Chapter: 6 Problem: 6.1.1.449

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use an algebraic method in Exercises 3944 to find the cube roots of the complex number. Approximate exact solution values when appropriate. \(2(\cos 2 \pi+i \sin 2 \pi)\)

Chapter: 6 Problem: 6.1.1.450

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use an algebraic method in Exercises 3944 to find the cube roots of the complex number. Approximate exact solution values when appropriate. \(2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\)

Chapter: 6 Problem: 6.1.1.451

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use an algebraic method in Exercises 3944 to find the cube roots of the complex number. Approximate exact solution values when appropriate. \(3\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)\)

Chapter: 6 Problem: 6.1.1.452

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use an algebraic method in Exercises 3944 to find the cube roots of the complex number. Approximate exact solution values when appropriate. \(27\left(\cos \frac{11 \pi}{6}+i \sin \frac{11 \pi}{6}\right)\)

Chapter: 6 Problem: 6.1.1.453

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use an algebraic method in Exercises 3944 to find the cube roots of the complex number. Approximate exact solution values when appropriate. \(3-4 i\)

Chapter: 6 Problem: 6.1.1.454

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Use an algebraic method in Exercises 3944 to find the cube roots of the complex number. Approximate exact solution values when appropriate. \(-2+2 i\)

Chapter: 6 Problem: 6.1.1.455

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4550, find the fifth roots of the complex number. \(\cos \pi+i \sin \pi\)

Chapter: 6 Problem: 6.1.1.456

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4550, find the fifth roots of the complex number. \(32\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\)

Chapter: 6 Problem: 6.1.1.457

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4550, find the fifth roots of the complex number. \(2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)\)

Chapter: 6 Problem: 6.1.1.458

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4550, find the fifth roots of the complex number. \(2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\)

Chapter: 6 Problem: 6.1.1.459

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4550, find the fifth roots of the complex number. \(2 i\)

Chapter: 6 Problem: 6.1.1.460

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4550, find the fifth roots of the complex number. \(1+\sqrt{3} i\)

Chapter: 6 Problem: 6.1.1.461

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5156, find the nth roots of the complex number for the specified value of n. \(1+i, \quad n=4\)

Chapter: 6 Problem: 6.1.1.462

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5156, find the nth roots of the complex number for the specified value of n. \(1-i, \quad n=6\)

Chapter: 6 Problem: 6.1.1.463

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5156, find the nth roots of the complex number for the specified value of n. \(2+2 i, \quad n=3\)

Chapter: 6 Problem: 6.1.1.464

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5156, find the nth roots of the complex number for the specified value of n. \(-2+2 i, \quad n=4\)

Chapter: 6 Problem: 6.1.1.465

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5156, find the nth roots of the complex number for the specified value of n. \(-2 i, \quad n=6\)

Chapter: 6 Problem: 6.1.1.466

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5156, find the nth roots of the complex number for the specified value of n. \(32, \quad n=5\)

Chapter: 6 Problem: 6.1.1.467

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5760, express the roots of unity in standard form \(a+b i\). Graph each root in the complex plane. Cube roots of unity

Chapter: 6 Problem: 6.1.1.468

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5760, express the roots of unity in standard form \(a+b i\). Graph each root in the complex plane. Fourth roots of unity

Chapter: 6 Problem: 6.1.1.469

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5760, express the roots of unity in standard form \(a+b i\). Graph each root in the complex plane. Sixth roots of unity

Chapter: 6 Problem: 6.1.1.470

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5760, express the roots of unity in standard form \(a+b i\). Graph each root in the complex plane. Square roots of unity

Chapter: 6 Problem: 6.1.1.471

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5760, express the roots of unity in standard form \(a+b i\). Graph each root in the complex plane. Determine z and the three cube roots of z if one cube root of z is \(1+\sqrt{3} i\).

Chapter: 6 Problem: 6.1.1.472

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 5760, express the roots of unity in standard form \(a+b i\). Graph each root in the complex plane. Determine z and the four fourth roots of z if one fourth root of z is -2 - 2i.

Chapter: 6 Problem: 6.1.1.473

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Quotient Formula Let \(z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)\) and \(z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right), r_{2} \neq 0\). Verify that \(z_{1} / z_{2}=r_{1} / r_{2}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]\)

Chapter: 6 Problem: 6.1.1.474

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activity nth Roots Show that the nth roots of the complex number \(r(\cos \theta+i \sin \theta)\) are spaced \(2 \pi / n\) radiants apart on a circle with radius \(\sqrt[n]{r}\).

Chapter: 6 Problem: 6.1.1.475

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The trigonometric form of a complex number is unique. Justify your answer.

Chapter: 6 Problem: 6.1.1.476

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The complex number i is a cube root of -i. Justify your answer.

Chapter: 6 Problem: 6.1.1.477

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6770, do not use technology to solve the problem. Multiple Choice Which of the following is a trigonometric form of the complex number \(-1+\sqrt{3} i\)? \(\begin{array}{l}\text{(A) } 2\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\\\text{(B) } 2\left(\cos \frac{2 \pi}{3}+i

Chapter: 6 Problem: 6.1.1.478

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6770, do not use technology to solve the problem. Multiple Choice Which of the following is the number of distinct complex number solutions of \(z^{5}=1+i\)? \(\begin{array}{l}\text{(A) 0}\\\text{(B) 1}\\\text{(C) 3}\\\text{(D) 4}\\\text{(E) 5}\end{array}\)

Chapter: 6 Problem: 6.1.1.479

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6770, do not use technology to solve the problem. Multiple Choice Which of the following is the standard form of the product of \(\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \text { and } \sqrt{2}\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)\)? \(\begi

Chapter: 6 Problem: 6.1.1.480

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 6770, do not use technology to solve the problem. Multiple Choice Which of the following is not a fourth root of 1? \(\begin{array}{l}\text{(A) } i^{2}\\\text{(B) } -i^{2}\\\text{(C) } \sqrt{-1}\\\text{(D) } -\sqrt{-1}\\\text{(E) } \sqrt{i}\end{array}\)

Chapter: 6 Problem: 6.1.1.481

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Complex Conjugates The complex conjugate of \(z=a+b i \text { is } \bar{z}=a-b i \text {. Let } z=r(\cos \theta+i \sin \theta)\). (a) Prove that \(\bar{z}=r[\cos (-\theta)+i \sin (-\theta)]\). (b) Use the trigonometric form to find \(z \cdot \bar{z}\). (c) Use the trigonometric form to fi

Chapter: 6 Problem: 6.1.1.482

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Modulus of Complex Numbers Let \(z=r(\cos \theta+i \sin \theta)\). (a) Prove that \(|z|=|r|\). (b) Use the trigonometric form for the complex numbers \(z_{1}\) and \(z_{2}\) to prove that \(\left|z_{1} \cdot z_{2}\right|=\left|z_{1}\right| \cdot\left|z_{2}\right|\).

Chapter: 6 Problem: 6.1.1.483

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Using Polar Form on a Graphing Calculator The complex number \(r(\cos \theta+i \sin \theta)\) can be entered in polar form on some graphing calculators as \(r e^{i \theta}\). (a) Support the result of Example 3 by entering the complex numbers \(z_{1} \text { and } z_{2}\) in polar form on your gr

Chapter: 6 Problem: 6.1.1.484

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Visualizing Roots of Unity Set your graphing calculator in parametric mode with \(0 \leq T \leq 8, \text { Tstep }=1\), \(X \min =-2.4, X \max =2.4, \mathrm{Ymin}=-1.6 \text {, and } \mathrm{Ymax}=1.6\). (a) Let \(x=\cos ((2 \pi / 8) t) \text { and } y=\sin ((2 \pi / 8) t)\). Use TRACE to visuali

Chapter: 6 Problem: 6.1.1.485

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Parametric Graphing Write parametric equations that represent \((\sqrt{2}+i)^{n} \text { for } n=t\) for Draw and label an accurate spiral representing \((\sqrt{2}+i)^{n}\) for n = 0, 1, 2, 3, 4.

Chapter: 6 Problem: 6.1.1.486

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Parametric Graphing Write parametric equations that represent \((-1+i)^{n} \text { for } n=t\). Draw and label an accurate spiral representing for \((-1+i)^{n} \text { for } n=0,1,2,3,4\).

Chapter: 6 Problem: 6.1.1.487

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Explain why the triangles formed by 0, 1, and \(z_{1}\), and by 0, \(z_{2}\), and \(z_{1} z_{2}\) shown in the figure are similar triangles.

Chapter: 6 Problem: 6.1.1.488

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Compass and Straightedge Construction Using only a compass and straightedge, construct the location of \(z_{1} z_{2}\) given the location of 0, 1, \(z_{1}, \text { and } z_{2}\).

Chapter: 6 Problem: 6.1.1.489

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7984, find all solutions of the equation (real and complex). \(x^{3}-1=0\)

Chapter: 6 Problem: 6.1.1.490

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7984, find all solutions of the equation (real and complex). \(x^{4}-1=0\)

Chapter: 6 Problem: 6.1.1.491

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7984, find all solutions of the equation (real and complex). \(x^{3}+1=0\)

Chapter: 6 Problem: 6.1.1.492

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7984, find all solutions of the equation (real and complex). \(x^{4}+1=0\)

Chapter: 6 Problem: 6.1.1.493

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7984, find all solutions of the equation (real and complex). \(x^{5}+1=0\)

Chapter: 6 Problem: 6.1.1.494

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 7984, find all solutions of the equation (real and complex). \(x^{5}-1=0\)

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